Equilibrium Bank Runs Revisited
Ed Nosal
Federal Reserve Bank of Chicago
August 2011
This version: September 2011
Abstract
In a Diamond and Dybvig (1983) environment, Green and Lin (2003) take a
mechanism design approach and show that a bank run equilibrium cannot exist.
Peck and Shell (2003) generalize their economic environment and show that it
can. The bank run, however, does not emerge because of modi…cations to
the economic environment but rather because the mechanism that implements
allocations is not an optimal one. When an optimal mechanism is used, the
Peck and Shell (2003) bank run equilibrium disappears.
1
Introduction
Even though Diamond and Dybvig’s (1983) seminal article is famously associated
with bank runs, bank runs ‘don’t come easy’ in their environment. For example,
in the …rst part of their article, where there is no aggregate risk, they demonstrate
that a bank run equilibrium cannot exist when the optimal deposit contract is in
place. In the second part of their article, where there is aggregate risk, their analysis
ignores the sequential service constraint, (Wallace (1988)). If there is no sequential
service constraint, then there is no bank run equilibrium associated with the optimal
deposit contract. Subsequently, Green and Lin (2003), GL, take a mechanism design
approach, fully account for the sequential service constraint and demonstrate that the
optimal bank contract does not admit a bank run equilibrium. Peck and Shell (2003),
PS, make two modi…cations to the GL model and produce a bank run equilibrium.
A few words about the modi…cations. First, PS alter the preferences of depositors
so that incentive constraints bind— incentive constraints do not bind for GL’s preferences. Second, PS assume that depositors do not know their positions in the service
queue— GL assume they do— which means that GL’s powerful backward induction
argument does not apply. If anything, PS’s modi…cations generalize the GL environment. PS adopt the preferences of Diamond and Dybvig (1983), where patient
1
and impatient depositors can have di¤erent utility functions. GL restrict these functions to be the same. If depositors do not know their positions in the service queue,
then the mechanism can choose to either inform or not inform depositors regarding
their positions. GL can be interpreted as restricting the mechanism to always inform
depositors about their positions in the queue.
Independent of how one views the GL environment vis-à-vis the PS environment,
GL use a mechanism that is optimal for their economic environment. Without loss in
generality, they restrict their attention to direct revelation mechanisms, where each
depositor makes an announcement regarding his private information or type. PS also
use a direct revelation mechanism. But for their more general economic environment, a direct revelation mechanism is not an optimal one. I construct an indirect
mechanism for the PS environment that uniquely implements the best implementable
allocation. In other words, the indirect mechanism does not admit a bank run equilibrium. This result reinforces an earlier observation: Bank runs are hard to come by
in the Diamond-Dybvig environment.
Cavalcanti and Monteiri (2011) examine the use of indirect mechanisms in a GL
environment. Their exercise is relevant because a bank run equilibrium can arise in a
GL environment when a direct mechanism is used and depositors’types are correlated,
see Ennis and Keister (2009).1 By expanding the message space for depositors and
appropriately designing ‘o¤ equilibrium’ payo¤s so that the planner can learn the
true type of each depositor, Cavalcanti and Monteiri (2011) demonstrate that the
best implementable allocation can be uniquely implemented by using a backward
induction argument. This backward induction argument, however, will not work for
the more general PS environment, where depositors do not know their positions in
the queue.2 The indirect mechanism proposed in this paper is su¢ ciently general to
uniquely implement the best implementable allocation for both the GL- and PS-type
environments.
The paper is organized as follows. The next section describes the economic environment. Section 3 characterizes the best implementable allocation. Sections 4 and 5
construct mechanisms that uniquely implements it. A concluding comment is o¤ered
in the …nal section.
2
Environment
There are three dates: 0, 1 and 2. The economy is endowed with Y > 0 units of
date-1 goods. A constant returns to scale technology transforms y units of date-1
goods into yR > y units of date-2 goods.
1
GL assume that depositor types are identically and independently distributed.
Cavalcanti and Monteiri (2011) propose an alternative indirect mechanism when they examine
a PS environment. In one example, they show that their indirect mechanism uniquely implements
the best implementable allocation. In another example, their indirect mechanism has a bank run
equilibrium.
2
2
There are N ex ante identical agents. An agent is one of two types t 2 T = f1; 2g:
patient, t = 1, or impatient, t = 2. The utility function for an impatient agent is
u (c1 ) and the utility function for a patient agent is v (c1 + c2 ), where c1 is date-1
consumption and c2 is date-2 consumption. u and v are increasing, strictly concave,
and twice continuously di¤erentiable. Agents maximize expected utility.
The number of patient agents in economy is drawn from the probability distribution = ( 0 ; : : : ; N ), where n > 0, n 2 f1; : : : N g
N, is the probability that
N
there are n patient agents. A queue is the vector t = (t1 ; : : : ; tN ) 2 T N , where
tk 2 T is the type of agent that occupies the k th position/coordinate in the queue.
Let Pn = tN 2 T N j#2 2 tN = n and Qn = jjtj = 2 for tN 2 Pn , where ‘#2’is
the number of patient agents. Pn is the set of queues with n patient agents and Qn is
the queue positions of the n patient agents in tN 2 Pn . The probability that tN 2 Pn
is n =#Pn = n = Nn , where #Pn is the number of queues tN 2 Pn . This speci…cation
implies that all potential queues with n patient agents are equally likely. Agents are
randomly assigned a position in the queue, where the (unconditional) probability that
an agent is assigned to position k is 1=N . For convenience, call the agent assigned to
position k agent k.
The queue realization, tN , is observed by no one: not by any of the agents nor
the planner. Each agent, however, privately observes his type t 2 T .
The timing of events and actions is as follows. At date 0, the planner constructs a
mechanism that determines how date-1 and date-2 consumption are allocated among
the N agents, and queue tN is realized. A mechanism is a set of announcements, M
and A, and a allocation rule, c = (c1 ; c2 ) where c1 = (c11 ; : : : c1N ) and c2 = (c21 ; : : : c2N ).
At date 1, agents sequentially meet the planner, starting with agent 1. In a meeting
with agent k, the planner announces ak 2 A and agent k responds with mk 2 M .
Only agent k and the planner can directly observe ak and mk . (But the planner
can reveal (ak ; mk ) to agent j
k via announcement aj , if he wishes.) There is
a sequential service constraint at date 1, which means the planner allocates date1 consumption to agent k 2 N based on the announcements of agents j
k, i.e.,
c1k mk 1 ; mk , where mk 1 = (m1 ; : : : ; mk 1 ). Agents consume the date-1 good at
their date-1 meetings with the planner. After all agents have met the planner, the
planner simultaneously allocates the date-2 consumption good to each agent based on
all of the date-1 announcements made by the agents, i.e., agent k receives c2k mN ,
where mN = (m1 ; : : : ; mN ) 2 M N .
3
Best Weakly Implementable Allocation
An allocation is weakly implementable is if it is an outcome to some equilibrium of
the mechanism; it is strongly (or uniquely) implementable if it is an outcome to every
equilibrium of the mechanism. Among the set of weakly implementably allocations,
the best weakly implementable allocation provides agents with the highest expected
utility. To characterize the best weakly implementable allocation, it is without loss of
3
generality to restrict the planner to use a direct revelation mechanism, where agents
make truthful announcement, mk = tk 2 M D = f1; 2g. The economy-wide welfare—
which is the expected utility of an agent before he learns his type— associated with
allocation rule c when agents use strategies mk 2 M D is
N
X X
n
U
N
N
n t 2Pn k=1
N
X
n=0
c1k mk 1 ; mk ; c2k mN
1 ; tk ;
(1)
where
1
k 1
U c1k mk 1 ; mk ; c2k mN
; mk
1 ; tk = u ck m
if tk = 1
and
k 1
1
; mk + c2k mN
U c1k mk 1 ; mk ; c2k mN
1 ; tk = v ck m
if tk = 2
The allocation rule c is feasible, i.e., there exists su¢ cient resources to pay for c
for all mk 2 M D , k 2 N, if
!
N
N
X
X
1
k 1
c2k mN :
(2)
ck m ; mk
R Y
k=1
k=1
Allocation rule c must be incentive compatible in the sense that agent k has no
reason to announce mk 6= tk . Since impatient agent k only values date-1 consumption,
he always announces mk = 1.3 When A = ;,4 patient agent k has no incentive to
depart from the strategy mk = 2, assuming that all other agents j announce mj = tj ,
if
N
X
^n
n=1
N
X
n=1
X 1 X
v c1k tk 1 ; 2 + c2k tk 1 ; 2; tN
k+1
n k2Q
N
t 2Pn
^n
X 1 X
v c1k tk 1 ; 1 + c2k tk 1 ; 1; tN
k+1
n
N
k2Q
t 2Pn
;
n
where xji = (xi ; : : : ; xj ) and
^ n = PN
N
n= n
n=1
3
(3)
n
N
n= n
This anticipates the result that the best weakly implementable allocation provides zero date-1
consumption to patient agents.
4
To characterize the best weakly implementable allocation, one wants to choose from the largest
possible set of incentive compatible allocations. This occurs when A = ;. In particular, when A = ;,
there is only one incentive compatibility constraint for all patient agents, (3). When A 6= ;, there
will be distinct incentive constraints for agents k who receive information ak from the mechanism.
Since an appropriately weighted average of these distinct incentive constraints reduces to the single
incentive constraint (3), the set of incentive compatible allocations when A 6= ; is a subset of the
set of incentive compatible allocations when A = ;.
4
is the conditional probability that agent k is in a speci…c queue that has n patient
agent. The 1=n terms that appear in (3) re‡ect that a patient agent has a 1=n chance
of occupying each of the patient queue positions in Qn .
The best weakly implementable allocation, denoted as c = (c1 ; c2 ), is given by
the solution to
max (1) subject to (2) and (3);
c
(4)
where mk = tk for all k 2 N in (1) and (2).
When agents use truth-telling strategies, c has the feature that impatient agents
consume at date 1 and patient agents consume only at date 2. The allocation rule c
corresponds to the analysis contained in PS’s Appendix B.
Both PS and Ennis and Keister (2009) demonstrate, by example, that mechanism
M D ; c can have two equilibria: one where agents play truth-telling strategies,
mk = tk for all k 2 N, and another where agents play bank run strategies, mk = 1
for all k 2 N.5 The bank run equilibria arise in these examples because the direct
revelation mechanism M D ; c is not an optimal mechanism. An optimal mechanism
may be a direct mechanism with A 6= ; or an indirect mechanism, (or both).
Direct Mechanisms with A 6= ;
4
When c cannot be uniquely implemented by the direct mechanism M D ; c , the optimal mechanism may be a direct mechanism with A 6= ;, i.e., A; M D ; c . Consider
…rst the example provided by Ennis and Keister (2009). Ennis and Keister (2009)
assume the preference speci…cation of GL, which implies that incentive constraint (3)
does not bind for allocation c . In addition, we know from GL that when A = N
and ak = k, i.e., the planner announces the agent’s position in the queue, none of
the N incentive compatibility constraints for patient agents bind. This means that
mechanism A = N; M D ; c can weakly implement the best allocation in c . And the
main result of GL implies that mechanism A = N; M D ; c can strongly implement
c . Therefore, A = N; M D ; c is an optimal mechanism; M D ; c admits a bank
run equilibrium only because it is a suboptimal mechanism.
Consider now the example provided by PS in their Appendix B. Nosal and Wallace
(2009) show that the best weakly implementable allocation, c , is not weakly implementable if the direct mechanism is characterized by A = N and M = f1; 2g. This
implies that the mechanism used by PS, M D ; c , is an optimal direct mechanism.
But the optimal mechanism may not be a direct mechanism.
5
The Ennis and Keister (2009) example that I refer to is their bank run example in section 4.2
of their paper, where agents do not know their position in the queue, as in PS, but where the utility
functions of patient and impatient agents are the same, as in GL.
5
5
Indirect Mechanisms
Suppose that mechanism M D ; c weakly, but not strongly, implements the best
allocation in c , and that mechanism A; M D ; c , where A 6= ;, cannot weakly implement the best allocation in c . Since a direct mechanism cannot uniquely implement
the best allocation in c , I construct an indirect mechanism that can.
The indirect mechanism M I ; c has M I 2 f1; 2; (2; r)g. The mechanism exploits
the fact that each agent knows when all other agents are playing bank run strategies.
One can interpret the message mk = (2; r) as agent k telling the planner he is patient
and other (patient) agents are playing bank run strategies. The mechanism rewards
agent k if evidence supports that notion that other agents are playing bank run
strategies, and punishes otherwise. The evidence supports a bank run if all agents
j 2 Nn fkg announce either mj = 1 or mj = (2; r), and does not support a bank
run if some agent j 2 Nn fkg announces mj = 2. For convenience of presentation, I
assume that agents are restricted to play pure strategies, (the appendix examines the
case where agents can play randomized strategies). Since agents do not know their
position in the queue, there are only two possible pure strategy equilibria: one where
agents play truth-telling strategies and another where they play bank run strategies.
The following notation is needed. The allocation rule for the indirect mechanism
is c = (c1 ; c2 ), and the date-s payo¤ to agent j who announces mj is denoted as csj jmj .
De…ne Z as the set of agents who announce (2; r), i.e., Z = (2; r) = fjjmj = (2; r)g.
De…ne m
^ k 1 as the message vector of length k 1 where for each j k 1, m
^j = 1
if either mj = 1 or mj = (2; r), and m
^ j = 2 if mj = 2.
I now specify the allocation rule c for the indirect mechanism M I ; c . If agent j
announces mj = (2; r), then
c1j j(2;r) = 0
c2j j(2;r) =
(5)
^ j 1 ; 1) (1 + ") if mi 2 f1; (2; r)g for all i 6= j;
c1j (m
0
if mi = 2 for some i 6= j;
where R > 1 + ", and " > 0. One can interpret "c1j (mj 1 ; 1) as a reward that agent
j receives for announcing mj = (2; r) when the evidence supports a bank run. If the
evidence does not support a bank run, then agent j receives zero. The mechanism
is able to gather evidence regarding bank runs since an agent who announces (2; r)
receives his consumption payment at date 2.
If agent j announces mj = 1, then
8 1
^ j 1 ; 1)
if j < N
< cj (m
1
1
cj (m
^ j 1 ; 1) +
if j = N , for all k; mk 6= 2
cj j1 =
;
(6)
: 1
j 1
cj (m
^ ; 1) + 2 if j = N , mk = 2 for some k
c2j j1 = 0:
Allocation rule (5) has the feature that if some agents announce mk = (2; r), then the
planner accumulates “excess goods.” That is, instead of making a date-1 payment
6
of c1 (m
^ j 1 ; 1) to an agent j that announces mj = (2; r), the planner gives either
1
j 1
c (m
^ ; 1) (1 + ") or zero at date 2, where R > 1 + ". If no agent j announces
mj = 2, then the total amount of this excess, denoted as , is
=
X (R
") c1z (m
^ z 1 ; 1)
;
R
1
z2Z
if some agent j announces mj = 2, then the total excess, denoted as
X
^ z 1; 1 :
c1z m
2 =
2,
is
z2Z
According to (6), agents who announce mk = 1 and occupy the …rst N 1 positions
in the queue receive the consumption payo¤ that they would get under the direct
^ k 1 is used as the announcement
revelation mechanism M D ; c , assuming that m
vector. Agent N who announces mN = 1 receives an additional consumption payment
of either or 2 .
Finally, if agent j announces mj = 2, then
c1j j2 = 0;
c2j j2 =
(7)
c2j
c2j
N
m
^
m
^N +
if j < N
:
R
if j = N
2
The structure of the payments associated with c2j resembles that of c1j . However,
does not appear in c2j because agent j announces mj = 2, and in this situation any
agent k who announces mk = (2; r) will receive zero. Note that the allocation rule
(5)-(7) has the planner sometimes throwing away goods. This happens when agent
N announces (2; r).
Proposition 1 The indirect mechanism M I ; c uniquely implements in pure strategies the best weakly implementable allocation in c .
Proof. First, there does not exist an equilibrium where all patient agents j announce
mj = 1. Suppose that such an equilibrium exists. Suppose that patient agent k defects from proposed play and announces mk = (2; r). Since all agents j 2 Nn fkg are
playing mj = 1 in the equilibrium, (5) speci…es a payment of c1k mk 1 ; 1 (1 + ") to
agent k which strictly exceeds the equilibrium payment of c1k mk 1 ; 1 ; a contradiction.
Second, there does not exist an equilibrium where all patient agents j announce
mj = (2; r). Suppose such an equilibrium exists. Then, the equilibrium expected
utility to patient agent k is
N
X
n=1
^n
X 1 X
f
v c1k
n
N
k2Q
t 2Pn
n
7
m
^ k 1 ; 1 (1 + ") g:
(8)
Suppose that patient agent k defects from the proposed equilibrium and announces
mk = 1. Using (6), his expected utility is
N
X
^n
n=1
X 1 X
f
v[c1k
n k2Q
N
t 2Pn
m
^ k 1; 1 +
X
(R
1
") c1j
m
^ j 1 ; 1 ];
(9)
j2Qn
j6=N
n
where
=
1 if k = N;
:
0 otherwise
Note that as " ! 0, the di¤erence between (9) and (8) is
N
X
n=2
^n
X 1 X
f
v[c1k
n
N
k2Q
t 2Pn
n
m
^ k 1; 1 +
X
(R
j2Qn
j6=N
1) c1j
m
^ j 1 ; 1 ] v c1k
m
^ k 1 ; 1 g > 0:
Hence, for any given N , , and R > 1, the mechanism can choose " > 0 su¢ ciently
small so that the value of (9) strictly exceeds that of (8), a contradiction.
Finally, consider a truth-telling equilibrium, where mj = tj for all j. Constraint
(3) implies that patient agent k does not have a strict incentive announce mk = 1. If
patient agent k strictly prefers to announce mk = 1 to mk = (2; r) when all agents
j 2 Nn fkg are playing mj = tj , then he has no incentive to announce mk = (2; r)
when all agents j 2 Nn fkg are playing mj = tj . Suppose patient agent k announces
mk = (2; r). If tN 2 P1 , i.e., agent k is the only patient agent in the queue, then
his consumption payment will be c1k m
^ k 1 ; 1 (1 + ), compared to c1k m
^ k 1 ; 1 if
N
he announces mk = 1. If, however, t 2 Pn , n 2— an event that occurs with strict
positive probability— then agent k’s consumption will be zero. For " > 0 arbitrarily
small, the expected utility associated with announcing mk = (2; r) is strictly less
than announcing mk = 1. Therefore, agent k strictly prefers to announce mk = 2 to
mk = (2; r).
All of this implies that when " > 0 is arbitrarily small, the best weakly implementable allocation in c is uniquely implemented by the mechanism M I ; c .
Agents do not know their positions in the queue for the indirect mechanism
M I ; c . Suppose that the economic environment is modi…ed so agents not only
learn their type, but they also (somehow) learn their position in the queue. Proposition 1 and its proof remains valid for the modi…ed economic environment, where
agents know their positions in the queue.6
6
Of course, the ‘c ’for the modi…ed environment may be di¤erent than the solution to (4). The
best weakly implementable allocation for the new environment is given by maxc (1) subject to (2)
and N incentive compatibility constraints, one for each patient agent in position j 2 N in the queue.
8
6
Final Comment
The insights from the …rst part of the Diamond and Dybvig (1983) paper are as
relevant today as they were almost 30 years ago: a poorly designed deposit contract
can invite bank runs, and a well designed one can prevent them.
7
Appendix: Randomized Strategies
Consider …rst a slightly modi…ed version of incentive constraint (3),
N
X
^n
n=1
N
X
n=1
^n
X 1 X
v c1k tk 1 ; 2 + c2k tk 1 ; 2; tN
k+1
n k2Q
N
t 2Pn
n
X 1 X
v c1k tk 1 ; 1 + c2k tk 1 ; 1; tN
k+1
n
N
k2Q
t 2Pn
(10)
+ ;
n
where
0. When = 0, (10) is identical to (3). If > 0, then any patient
agent j strictly prefers to announce mj = 2 to mj = 1, assuming all other agents
i 2 Nn fjg announce mi = ti . The best -weakly implementable allocation, denoted
by c ( ) = (c1 ( ) ; c2 ( )), is given by the solution to
max (1) subject to (2) and (10),
c( )
where mk = tk for all k 2 N in (1) and (2).
Note that as ! 0, c ( ) ! c and c (0) c .
Agents can play randomized strategies. Denote the indirect mechanism as M I ; cI ,
where M I 2 f1; 2; (2; r)g and cI = c1I ; c2I . The basic structure of the allocation
rule cI is similar to c presented in text, (5)-(7), but the construction of cI uses c ( )
and not c (0) = c . To reduce notational clutter I will suppress the ‘ ’when using
allocations in c ( ) to describe cI . If agent j announces mj = (2; r), then
c1I
j j(2;r) = 0;
1
c2I
j j(2;r) = cj
m
^ j 1 ; 1 (1 + ") for all j; " > 0:
Note that, unlike c2j j(2;r) in the text, (5), agents are never “penalized”for announcing
mj = (2; r). If agent j announces mj = 1, then
c1I
j j1 =
c1j (m
^ j 1 ; 1)
c1j (m
^ j 1 ; 1) +
c2I
j j1 = 0:
9
if j < N
;
if j = N
Since agents are not penalized for announcing (2; r), the 2 term is now irrelevant
1
and does not appear in c1I
j j1 , (compared to cj j1 in the text, (6).) Finally, if agent j
announces mj = 2, then
c1I
j j2 = 0;
c2I
j j2 =
c2j
c2j
m
^N
m
^N +
if j < N
:
R if j = N
Here, as in the text, (7), an agent who announces mj = 2, where j = N , will collect
an additional payment, R, where R > 0 if Z 6= ;.
Proposition 2 The indirect mechanism M I ; cI uniquely implements an allocation
that is arbitrarily close to the best weakly implementable allocation in c .
Proof. First, there does not exist an equilibrium where patient agents j announce
mj = 2 with probability 1 and mj = 1 with probability 1
1 , where
1 < 1.
Suppose that such an equilibrium exists. Suppose that patient agent k defects from
proposed play and announces mk = (2; r) with probability one. His consumption will
be c1j (m
^ j 1 ; 1) (1 + ") which is strictly greater than the equilibrium consumption
j 1
1
^ ; 1); a contradiction.
cj (m
Second, there does not exist an equilibrium where patient agents j announce
mj = 1 with probability 2 and mj = (2; r) with probability 1
2 , where 2 < 1.
(The case where 2 = 1 is covered in the …rst step, above.) Suppose that such an
equilibrium exists. Suppose that patient agent k defects from proposed play and
^ j 1 ; 1) (1 + ")
announces mk = 1 with probability one. Note that as " ! 0, c1j (m
^ j 1 ; 1) ! 0 for all j < N . Conditional on patient agent j not occupying the
c1j (m
th
N position in the queue, the di¤erence in the expected utility of patient agent j
announcing mj = 1 and announcing mj = (2; r), by continuity, tends to zero as
" ! 0. Patient agent j = N receives a consumption payo¤ of c1j (m
^ j 1 ; 1) + if he
1
j 1
announces mj = 1 and cj (m
^ ; 1) (1 + ") if he announces mj = (2; r). Hence, as
" ! 0,
c1j
m
^ j 1; 1 +
c1j
m
^ j 1 ; 1 (1 + ") !
X (R
z2Z
1) c1z (m
^ z 1 ; 1)
> 0 if Z 6= ;:
R
In the proposed equilibrium, the event Z 6= ; occurs with a probability that is
strictly greater than zero and minZ6=; f g is not “arbitrarily small.” For any given
N , , R and 2 < 1, the mechanism can choose an " > 0 su¢ ciently small so that
the expected utility of patient agent k announcing mk = 1 strictly exceeds that of
announcing mk = (2; r); a contradiction. This discussion implies that patient agent
k can never be indi¤erent between announcing mk = 1 and mk = (2; r) when " > 0 is
arbitrarily small; he strictly prefers to announce mk = 1 if other agents j announce
mj = (2; r) with positive probability.
10
Third, suppose there is an equilibrium where patient agents announce mj = 2 with
probability 3 and mj = (2; r) with probability 1
3 , where 3 < 1. Since patient
agents are asked to randomize in the equilibrium, it must be the case that patient
agent k is indi¤erent between announcing mk = 2 and mk = (2; r). But the above
discussion, in the second step, implies that patient agent k will defect from proposed
equilibrium play and announce mk = 1 with probability one, since he strictly prefers
announcing mk = 1 to mk = (2; r) (or mi = 2); a contradiction. There also cannot be
an equilibrium where patient agents j randomize over announcing mj = 1, mj = 2,
and mj (2; r); a patient agent k strictly prefers announcing mk = 1 with probability
one. The …rst three steps imply that the indirect mechanism does not admit equilibria
with randomization.
Finally, consider an equilibrium where agents of type tj announce mj = tj . Suppose that > 0. Then patient agent j strictly prefers announcing mj = 2 to mj = 1.
Patient agent k strictly prefers announcing mk = (2; r) to mk = 1 when all other
agents j 2 Nn fkg announce mj = tj . But for any > 0, there exists an " > 0 su¢ ciently small so that patient agent k strictly prefers announcing mk = 2 to mk = (2; r)
since > 0 implies that agent k strictly prefers announcing mk = 2 to mk = 1. Therefore, for > 0 arbitrarily small, c ( ) c , and the unique equilibrium for mechanism
M I ; cI is characterized by mj = tj for all j 2 N.
References
[1] Cavalcanti, Ricardo, de O., and Monteiri, Paulo, K. “Enriching Information to
Prevent Bank Runs.”manuscript, February 2011.
[2] Diamond, Douglas, and Dybvig, Phillip. “Bank Runs, Deposit Insurance, and
Liquidity.”J.P.E. 91 (June 1983): 401-19.
[3] Ennis, Huberto, M., and Keister, Todd. “Run Equilibria in the Green-Lin Model
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